Question

integration of e raise to power x ( sin4x - 4 / 1 - cos4x) DX
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Answer 1

Answer:

the integrand is of the form of e^x f(x) where

f(x) = (sin4x - 4)/(1 - cos4x)

though we can proceed to solve the integration as it is, we will have to do the things which we can obviate if we rely on a property of integrand involving e^x, the property is

integration of e^x (f(x) + f'(x)) dx = e^x f(x) + c

so I will proceed by relying on above property.

first we will see what is f(x). for this we proceed with

(sin4x - 4)/(1 - cos4x)

= (2sin2x cos2x - 4)/2 sin^2 (2x))

= 2sin2x cos2x / 2sin^2(2x) - 4/2sin^2(2x)

= cos2x/sin2x - 2/sin^2(2x)

= cot2x - 2cosec^2(2x)

let f(x) = cot2x then

f'(x) = -2cosec^2(2x)

thus our integrand is now

e^x ( cot2x + (-2cosec^2(2x)) dx

f(x). f'(x)

the solution will be

e^x f(x) + c

= e^x cot2x + c